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This book is the first attempt to develop systematically a general theory of the initial-boundary value problems for nonlinear evolution equations with pseudodifferential operators Ku on a half-line or on a segment. We study traditionally important problems, such as local and global existence of solutions and their properties, in particular much attention is drawn to the asymptotic behavior of solutions for large time. Up to now the theory of nonlinear initial-boundary value problems with a general pseudodifferential operator has not been well developed due to its difficulty. There are many open natural questions. Firstly how many boundary data should we pose on the initial-boundary value problems for its correct solvability? As far as we know there are few results in the case of nonlinear nonlocal equations. The methods developed in this book are applicable to a wide class of dispersive and dissipative nonlinear equations, both local and nonlocal. For the first time the definition of pseudodifferential operator on a half-line and a segment is done A wide class of nonlinear nonlocal and local equations is considered Developed theory is general and applicable to different equations The book is written clearly, many examples are considered Asymptotic formulas can be used for numerical computations by engineers and physicists The authors are recognized experts in the nonlinear wave phenomena.
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Nonlinear Evolution Equation presents state-of-the-art theories and results on nonlinear evolution equation, showing related mathematical methods and applications. The basic concepts and research methods of infinite dimensional dynamical systems are discussed in detail. The unique combination of mathematical rigor and physical background makes this work an essential reference for researchers and students in applied mathematics and physics.
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This book contains the papers presented at the ICM2002 Satellite Conference on Nonlinear Evolution Equations and Dynamical Systems. About 50 mathematicians and scientists attended the meeting - including E Witten (IAS), C Nappi (Princeton), K Khanin (Cambridge), D Phong (Columbia), d'Hoker (UCLA) and Peng Chiakuei (CAS). The book covers several fields, such as nonlinear evolution equations and integrable systems, infinite-dimensional algebra, conformal field theory and geometry. The proceedings have been selected for coverage in:. Index to Scientific & Technical Proceedings (ISTP CDROM versi
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Reporting a novel breakthrough in the identification and investigation of solvable and integrable nonlinearly coupled evolution ordinary differential equations (ODEs) or partial differential equations (PDEs), this text includes practical examples throughout to illustrate the theoretical concepts. Beginning with systems of ODEs, including second-order ODEs of Newtonian type, it then discusses systems of PDEs, and systems evolving in discrete time. It reports a novel, differential algorithm which can be used to evaluate all the zeros of a generic polynomial of arbitrary degree: a remarkable development of a fundamental mathematical problem with a long history. The book will be of interest to applied mathematicians and mathematical physicists working in the area of integrable and solvable non-linear evolution equations; it can also be used as supplementary reading material for general applied mathematics or mathematical physics courses.
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Bilinear transformation method
Benjamin-Ono equations. --- Bilinear transformation method. --- Evolution equations, Nonlinear --- Numerical solutions. --- Bilinearization of nonlinear evolution equations --- Transformation method, Bilinear --- B-O equations --- Numerical analysis --- Numerical solutions --- Bilinear transformation method --- Benjamin-Ono equations --- Evolution equations, Nonlinear - Numerical solutions
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Approximation of nonlinear evolution systems
Approximation theory. --- Evolution equations, Nonlinear --- Numerical solutions. --- Theory of approximation --- Functional analysis --- Functions --- Polynomials --- Chebyshev systems --- Numerical analysis
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Starting from physical motivations and leading to practical applications, this book provides an interdisciplinary perspective on the cutting edge of ultrametric pseudodifferential equations. It shows the ways in which these equations link different fields including mathematics, engineering, and geophysics. In particular, the authors provide a detailed explanation of the geophysical applications of p-adic diffusion equations, useful when modeling the flows of liquids through porous rock. p-adic wavelets theory and p-adic pseudodifferential equations are also presented, along with their connections to mathematical physics, representation theory, the physics of disordered systems, probability, number theory, and p-adic dynamical systems. Material that was previously spread across many articles in journals of many different fields is brought together here, including recent work on the van der Put series technique. This book provides an excellent snapshot of the fascinating field of ultrametric pseudodifferential equations, including their emerging applications and currently unsolved problems.
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This book mainly serves as an elementary, self-contained introduction to several important aspects of the theory of global solutions to initial value problems for nonlinear evolution equations. The book employs the classical method of continuation of local solutions with the help of a priori estimates obtained for small data. The existence and uniqueness of small, smooth solutions that are defined for all values of the time parameter are investigated. Moreover, the asymptotic behavior of the solutions is described as time tends to infinity. The methods for nonlinear wave equations are discussed in detail. Other examples include the equations of elasticity, heat equations, the equations of thermoelasticity, Schrödinger equations, Klein-Gordon equations, Maxwell equations and plate equations. To emphasize the importance of studying the conditions under which small data problems offer global solutions, some blow-up results are briefly described. Moreover, the prospects for corresponding initial-boundary value problems and for open questions are provided. In this second edition, initial-boundary value problems in waveguides are additionally considered.
Mathematics. --- Partial Differential Equations. --- Differential equations, partial. --- Mathématiques --- Initial value problems --- Evolution equations, Nonlinear --- Calculus --- Mathematics --- Physical Sciences & Mathematics --- Initial value problems. --- Evolution equations, Nonlinear. --- Nonlinear equations of evolution --- Nonlinear evolution equations --- Problems, Initial value --- Partial differential equations. --- Differential equations, Nonlinear --- Boundary value problems --- Differential equations --- Partial differential equations
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Evolution equations --- Evolution equations, Nonlinear --- Evolution equations. --- Equations, Evolution --- Equations of evolution --- Evolutionary equations --- Differential equations --- Mathematical Sciences --- Physics --- Algebra --- Algebraic Geometry --- Applied Mathematics --- Calculus --- Geometry --- Stochastic Computation --- General and Others
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